| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Sector and arc length |
| Difficulty | Moderate -0.3 This is a straightforward sector/arc length problem requiring standard formulas. Part (i) involves setting up an equation using arc lengths (s = rθ) with given perimeter, then solving for α. Part (ii) uses the sector area formula with the found angle. Both parts are routine applications of memorized formulas with minimal problem-solving, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \(2r\alpha + r\alpha + 2r = 4.4r\) | M1 | At least 3 of the 4 terms required |
| \(\alpha = 0.8\) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2}(2r)^2(0.8) - \frac{1}{2}(r^2)(0.8) = 30\) | M1A1\(\checkmark\) | Ft through on *their* \(\alpha\) |
| \((3/2)r^2 \times 0.8 = 30 \rightarrow r = 5\) | A1 [3] |
## Question 3:
### Part (i):
$2r\alpha + r\alpha + 2r = 4.4r$ | **M1** | At least 3 of the 4 terms required
$\alpha = 0.8$ | **A1** [2] |
### Part (ii):
$\frac{1}{2}(2r)^2(0.8) - \frac{1}{2}(r^2)(0.8) = 30$ | **M1A1**$\checkmark$ | Ft through on *their* $\alpha$
$(3/2)r^2 \times 0.8 = 30 \rightarrow r = 5$ | **A1** [3] |
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In the diagram $O C A$ and $O D B$ are radii of a circle with centre $O$ and radius $2 r \mathrm {~cm}$. Angle $A O B = \alpha$ radians. $C D$ and $A B$ are arcs of circles with centre $O$ and radii $r \mathrm {~cm}$ and $2 r \mathrm {~cm}$ respectively. The perimeter of the shaded region $A B D C$ is $4.4 r \mathrm {~cm}$.\\
(i) Find the value of $\alpha$.\\
(ii) It is given that the area of the shaded region is $30 \mathrm {~cm} ^ { 2 }$. Find the value of $r$.\\
$4 C$ is the mid-point of the line joining $A ( 14 , - 7 )$ to $B ( - 6,3 )$. The line through $C$ perpendicular to $A B$ crosses the $y$-axis at $D$.\\
\hfill \mbox{\textit{CAIE P1 2016 Q3 [5]}}